Question

Find the equation to the straight line which bisects the distance between the points $(a,b)$ and $(a^{\prime },b^{\prime })$ and also bisects the distance between the points $(-a,b)$ and $(a^{\prime },-b^{\prime })$.

Answer 1

The points which bisects the given points are $(x_1,y_1)=(\frac{a+a^{{}^{\prime }}}{2},\frac{b+b^{{}^{\prime }}}{2})$ and $(x_2,y_2)=(\frac{-a+a^{{}^{\prime }}}{2},\frac{b-b^{{}^{\prime }}}{2})$.......(1)

So we know point-point make an equation

$(y-y_1)=\frac{y_2-y_1}{x_2-x_1}\times (x-x_1)$.........(2)

Therefore substitute the above points (obtained in (1)) in the above equation (2)

We get the final equation on simplification to be

$\frac{y_2-y_1}{x_2-x_1}=\frac{\frac{b-b^{{}^{\prime }}-b-b^{{}^{\prime }}}{2}}{\frac{-a+a^{{}^{\prime }}-a-a^{{}^{\prime }}}{2}}=\frac{b^{{}^{\prime }}}{a}$

Then final equation on simplification by substituting all values is=

$2ay-2b^{{}^{\prime }}x-ab+a^{{}^{\prime }}b{}^{\prime }=0$